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Theorem ltbtwnnq 8597
Description: There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ltbtwnnq  |-  ( A 
<Q  B  <->  E. x ( A 
<Q  x  /\  x  <Q  B ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem ltbtwnnq
StepHypRef Expression
1 ltrelnq 8545 . . . . 5  |-  <Q  C_  ( Q.  X.  Q. )
21brel 4736 . . . 4  |-  ( A 
<Q  B  ->  ( A  e.  Q.  /\  B  e.  Q. ) )
32simprd 451 . . 3  |-  ( A 
<Q  B  ->  B  e. 
Q. )
4 ltexnq 8594 . . . 4  |-  ( B  e.  Q.  ->  ( A  <Q  B  <->  E. y
( A  +Q  y
)  =  B ) )
5 eleq1 2344 . . . . . . . . . 10  |-  ( ( A  +Q  y )  =  B  ->  (
( A  +Q  y
)  e.  Q.  <->  B  e.  Q. ) )
65biimparc 475 . . . . . . . . 9  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( A  +Q  y )  e.  Q. )
7 addnqf 8567 . . . . . . . . . . 11  |-  +Q  :
( Q.  X.  Q. )
--> Q.
87fdmi 5359 . . . . . . . . . 10  |-  dom  +Q  =  ( Q.  X.  Q. )
9 0nnq 8543 . . . . . . . . . 10  |-  -.  (/)  e.  Q.
108, 9ndmovrcl 5967 . . . . . . . . 9  |-  ( ( A  +Q  y )  e.  Q.  ->  ( A  e.  Q.  /\  y  e.  Q. ) )
116, 10syl 17 . . . . . . . 8  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( A  e. 
Q.  /\  y  e.  Q. ) )
1211simprd 451 . . . . . . 7  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  y  e.  Q. )
13 nsmallnq 8596 . . . . . . . 8  |-  ( y  e.  Q.  ->  E. z 
z  <Q  y )
1411simpld 447 . . . . . . . . . . . 12  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  A  e.  Q. )
151brel 4736 . . . . . . . . . . . . 13  |-  ( z 
<Q  y  ->  ( z  e.  Q.  /\  y  e.  Q. ) )
1615simpld 447 . . . . . . . . . . . 12  |-  ( z 
<Q  y  ->  z  e. 
Q. )
17 ltaddnq 8593 . . . . . . . . . . . 12  |-  ( ( A  e.  Q.  /\  z  e.  Q. )  ->  A  <Q  ( A  +Q  z ) )
1814, 16, 17syl2an 465 . . . . . . . . . . 11  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  A  <Q  ( A  +Q  z ) )
19 ltanq 8590 . . . . . . . . . . . . . 14  |-  ( A  e.  Q.  ->  (
z  <Q  y  <->  ( A  +Q  z )  <Q  ( A  +Q  y ) ) )
2019biimpa 472 . . . . . . . . . . . . 13  |-  ( ( A  e.  Q.  /\  z  <Q  y )  -> 
( A  +Q  z
)  <Q  ( A  +Q  y ) )
2114, 20sylan 459 . . . . . . . . . . . 12  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  ( A  +Q  z )  <Q  ( A  +Q  y ) )
22 simplr 734 . . . . . . . . . . . 12  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  ( A  +Q  y )  =  B )
2321, 22breqtrd 4048 . . . . . . . . . . 11  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  ( A  +Q  z )  <Q  B )
24 ovex 5844 . . . . . . . . . . . 12  |-  ( A  +Q  z )  e. 
_V
25 breq2 4028 . . . . . . . . . . . . 13  |-  ( x  =  ( A  +Q  z )  ->  ( A  <Q  x  <->  A  <Q  ( A  +Q  z ) ) )
26 breq1 4027 . . . . . . . . . . . . 13  |-  ( x  =  ( A  +Q  z )  ->  (
x  <Q  B  <->  ( A  +Q  z )  <Q  B ) )
2725, 26anbi12d 694 . . . . . . . . . . . 12  |-  ( x  =  ( A  +Q  z )  ->  (
( A  <Q  x  /\  x  <Q  B )  <-> 
( A  <Q  ( A  +Q  z )  /\  ( A  +Q  z
)  <Q  B ) ) )
2824, 27spcev 2876 . . . . . . . . . . 11  |-  ( ( A  <Q  ( A  +Q  z )  /\  ( A  +Q  z )  <Q  B )  ->  E. x
( A  <Q  x  /\  x  <Q  B ) )
2918, 23, 28syl2anc 645 . . . . . . . . . 10  |-  ( ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  /\  z  <Q  y
)  ->  E. x
( A  <Q  x  /\  x  <Q  B ) )
3029ex 425 . . . . . . . . 9  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( z  <Q 
y  ->  E. x
( A  <Q  x  /\  x  <Q  B ) ) )
3130exlimdv 1668 . . . . . . . 8  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( E. z 
z  <Q  y  ->  E. x
( A  <Q  x  /\  x  <Q  B ) ) )
3213, 31syl5 30 . . . . . . 7  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  ( y  e. 
Q.  ->  E. x ( A 
<Q  x  /\  x  <Q  B ) ) )
3312, 32mpd 16 . . . . . 6  |-  ( ( B  e.  Q.  /\  ( A  +Q  y
)  =  B )  ->  E. x ( A 
<Q  x  /\  x  <Q  B ) )
3433ex 425 . . . . 5  |-  ( B  e.  Q.  ->  (
( A  +Q  y
)  =  B  ->  E. x ( A  <Q  x  /\  x  <Q  B ) ) )
3534exlimdv 1668 . . . 4  |-  ( B  e.  Q.  ->  ( E. y ( A  +Q  y )  =  B  ->  E. x ( A 
<Q  x  /\  x  <Q  B ) ) )
364, 35sylbid 208 . . 3  |-  ( B  e.  Q.  ->  ( A  <Q  B  ->  E. x
( A  <Q  x  /\  x  <Q  B ) ) )
373, 36mpcom 34 . 2  |-  ( A 
<Q  B  ->  E. x
( A  <Q  x  /\  x  <Q  B ) )
38 ltsonq 8588 . . . 4  |-  <Q  Or  Q.
3938, 1sotri 5069 . . 3  |-  ( ( A  <Q  x  /\  x  <Q  B )  ->  A  <Q  B )
4039exlimiv 1670 . 2  |-  ( E. x ( A  <Q  x  /\  x  <Q  B )  ->  A  <Q  B )
4137, 40impbii 182 1  |-  ( A 
<Q  B  <->  E. x ( A 
<Q  x  /\  x  <Q  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1533    = wceq 1628    e. wcel 1688   class class class wbr 4024    X. cxp 4686  (class class class)co 5819   Q.cnq 8469    +Q cplq 8472    <Q cltq 8475
This theorem is referenced by:  nqpr  8633  reclem2pr  8667
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-recs 6383  df-rdg 6418  df-1o 6474  df-oadd 6478  df-omul 6479  df-er 6655  df-ni 8491  df-pli 8492  df-mi 8493  df-lti 8494  df-plpq 8527  df-mpq 8528  df-ltpq 8529  df-enq 8530  df-nq 8531  df-erq 8532  df-plq 8533  df-mq 8534  df-1nq 8535  df-rq 8536  df-ltnq 8537
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